# Universal Bounds and Monotonicity Properties of Ratios of Hermite and Parabolic Cylinder Functions

## TL;DR

This paper establishes universal bounds and monotonicity properties for ratios of Hermite and parabolic cylinder functions using probabilistic methods, providing new insights and Turán-type inequalities.

## Contribution

It introduces a novel probabilistic approach to prove monotonicity and bounds of function ratios, linking special functions with stochastic process eigenfunctions.

## Key findings

- Ratios are strictly decreasing and bounded by universal constants.
- Probabilistic methods effectively derive properties of special functions.
- New Turán-type inequalities are established.

## Abstract

We obtain so far unproved properties of a ratio involving a class of Hermite and parabolic cylinder functions. Those ratios are shown to be strictly decreasing and bounded by universal constants. Differently to usual analytic approaches, we employ simple purely probabilistic arguments to derive our results. In particular, we exploit the relation between Hermite and parabolic cylinder functions and the eigenfunctions of the infinitesimal generator of the Ornstein-Uhlenbeck process. As a byproduct, we obtain Tur\'an type inequalities.